Mr. Biden, whose team helped hammer out the deal, urged support for it on Monday in a statement from Karine Jean-Pierre, the White House press secretary, that said, “We strongly support this legislation and call on every senator to put partisan politics aside and vote to secure the border.”

Among other changes to immigration law, the measure would make it more difficult to gain asylum in the United States and increase detentions and deportations of those crossing into the country without authorization. It would also effectively close the border altogether if the average number of migrants encountered by immigration officials exceeded a certain threshold — an average of 5,000 over the course of a week or 8,500 on any given day. The bill also would give the president power to close the border unilaterally if migrant encounters reach an average of 4,000 per day over a week.

A cap on asylum is a violation of international law.